Where are they? If they don't exist, a secondary question: are there any reasons that of all the middle eastern religions of a few thousand years ago, why was Judaism the most well documented?
EDITED TO ADD I was more interested in "Old" Testament era. Yes, that is also a wide time period, but I am aware (enough) of the many books by Bart Ehrman and Elaine Pagels that treat Gnosticism and all the "lost christianities".
I'm not aware of similar accounts of, say, the Canaanaite religions, or the worship of Baal, or all those other rival deities that pre- and post-Exodus Jews were warned against. They are especially interesting to me because, unlike the early orthodox response to heretical Christianity, the Old Testament never says something along the lines of, "No, your god doesn't exist", or "Your interpretation of God is wrong"; rather, it's "Don't worship the dark gods" or "Yahweh is better at lighting a pile of wood on fire than your god."
Was there a particular religion or period that you had in mind? The bible obviously covers a very large period of history and many different religions and groups are mentioned across the old and new testaments.
Additionally there are internal groups within Judaism (especially after the destruction of the second temple in 70 AD) which could be mistaken rivaling religions. For example Jesus' criticism of the Pharisees in the New Testament.
Yes! Apollonius of Tyana. While this might not exactly answer your question, hopefully my answer gives you an idea about how religions competed for followers in the third and fourth centuries.
Apollonius of Tyana was written by a man named Philostratus who was commissioned to write the work as an encomium, i.e. propaganda, by Julia Domna mother of the Roman Emperor Caracalla. The work was commissioned to be the final say on Apollonius since there were a number of other works about him circulating at the time. Philostratus himself tells you to ignore book by Moiragenes on Apollonius, since it is his sources that are legit, not the other guys. Taking into account his sources, patron, and own words, it can be concluded that the “the point of Philostratus’ portrayal is to show the coherence between his philosophy, morals, and miraculous powers” and thus create a “propagandistic gospel intended to promote a following.”
This work presents its protagonist,Apollonius, as a divine man with a miraculous birth, precocious childhood, miracles in adult life, and divine death moments. He was positioned within the Greco-Roman world at this time as a rival to early Christianity and other divine-man religions.
Some of his miracles include:
ending a famine in Aspendus while under a vow of silence
tutoring the King of Babylon on how to run his kingdom according to his Greek philosophy
reforming temple rituals in a manner that further purified the rites
besting Emperor Domitian in a public trial
exorcism on an Indian boy(he traveled to Indian in the works)
healing a maimed youth by merely laying his hands on the injuries
returning the strength of a man who was paralyzed.
he had heard about the plague in Ephesus and being some many miles away, was instantly transported there. He proceeded to cure the city by ordering the citizens to stone a demon.
When in Rome he happened on a funeral of a girl and after touching her and whispering to her she came alive again.
Sources: The Stories themsevles: http://www.livius.org/ap-ark/apollonius/life/va_2_16.html#§16
Michael L. White, From Jesus to Christianity, San Francisco: HarperCollins, 2004.
There are lots, but I'll mention one which I don't think has been mentioned before. One of the early "rival" Jewish communities was the community at Elephantine in Egypt. Jeremiah 44 contains a list of correspondences between the prophet and the community in which he urges them to adopt a monotheistic religion and they refuse.
From the Egyptian side we not only have the responses contained in Jeremiah, but we also have the elephantine papyri, which indicates that the community worshipped a syncretistic polytheist form of the religion, including worship of joint deities like Anat-Yahu (fusion of the Canaanite/Ugaritic female warrior deity Anat with the Jewish Yahweh) and Anat-Baal (fusion of Anat with the Canaanite storm god Baal) and many others.
As mrlex wrote, there are numerous examples from numerous time periods covered by "the Bible," a collection of numerous documents.
I can give a few examples if that would be helpful.
The Genesis creation account is considered by some to be the Israelite answer to texts like the Enuma Elish and Atrahasis (Peter Enns is a very accessible source, either The Evolution of Adam, or Inspiration and Incarnation, can't remember which).
We also have some competing accounts of the wars of the Israelites; Sennacherib's Annals for example tell the other side of the story of Hezekiah's war with the Assyrians, recorded in 2 Kings 18.
The ancient Israelites almost certainly preserved some of the wisdom literature of the surrounding cultures within their own; I believe I saw these examples in Longman's commentary on Proverbs, along with a few interesting parallels from these other cultures.
As for Ushi007's follow up question, if we consider the Pharisees to be "rivals" in the New Testament, they are the most obvious candidate as predecessors to Rabbinic Judaism, whose earliest recorded document is the Mishnah, c. 200 C.E. Jacob Neusner has several introductory books which address the formation of Rabbinic Judaism.
The Sadducees on the other hand appear to have left no records--at least, none that we have discovered.
The Dead Sea Scrolls and the Old Testament Pseudepigrapha could also be considered documentation of "rival" Judaisms depending on how we construe their authors' relationship with "normative" Judaism... whatever that might have been at the time.
There are a number of texts from Ugarit that document portions of a Canaanite religion, including some cultic practices and foundational legends, such as the Ba‘al cycle and the epics of Danel (thus) and King Keret. I really cannot speak authoritatively on them—not my period—but I hope a specialist will chime in.